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Proceedings of the John H. Barrett Memorial Lectures held at the University of Tennessee, Knoxville, May 29–June 1, 2018
A Mathematical Introduction

## Abstract

Let 𝕋4 = 1, ±i} be the subgroup of 4-th roots of unity inside 𝕋, the multiplicative group of complex units. A complex unit gain graph Φ is a simple graph Γ = (V (Γ) = {v 1, . . . , vn}, E(Γ)) equipped with a map $ϕ:E→(Γ)→𝕋$ defined on the set of oriented edges such that ϕ(vivj) = ϕ(vjvi)−1. The gain graph Φ is said to be balanced if for every cycle C = vi 1 vi 2 vi k vi 1 we have ϕ(vi 1 vi 2)ϕ(vi 2 vi 3) ϕ(vi k vi 1) = 1.

It is known that Φ is balanced if and only if the least Laplacian eigenvalue λn(Φ) is 0. Here we show that, if Φ is unbalanced and ϕ(Φ) ⊆ 𝕋4, the eigenvalue λn(Φ) measures how far is Φ from being balanced. More precisely, let ν(Φ) (respectively, (Φ)) be the number of vertices (respectively, edges) to cancel in order to get a balanced gain subgraph. We show that

λn(Φ) ≤ ν(Φ) ≤ ∈(Φ).

We also analyze the case when λn(Φ) = ν(Φ). In fact, we identify the structural conditions on Φ that lead to such equality.

## Abstract

For vertex- and edge-connectivity we construct infinitely many pairs of regular graphs with the same spectrum, but with different connectivity.

## Abstract

The complementary spectrum of a connected graph G is the set of the complementary eigenvalues of the adjacency matrix of G. In this note, we discuss the possibility of representing G using this spectrum. On one hand, we give evidence that this spectrum distinguishes more graphs than other standard graph spectra. On the other hand, we show that it is hard to compute the complementary spectrum. In particular, we see that computing the complementary spectrum is equivalent to finding all connected induced subgraphs.

## Abstract

A nut graph is a singular graph with one-dimensional kernel and corresponding eigenvector with no zero elements. The problem of determining the orders n for which d-regular nut graphs exist was recently posed by Gauci, Pisanski and Sciriha. These orders are known for d ≤ 4. Here we solve the problem for all remaining cases d ≤ 11 and determine the complete lists of all d-regular nut graphs of order n for small values of d and n. The existence or non-existence of small regular nut graphs is determined by a computer search. The main tool is a construction that produces, for any d-regular nut graph of order n, another d-regular nut graph of order n+2d. If we are given a sufficient number of d-regular nut graphs of consecutive orders, called seed graphs, this construction may be applied in such a way that the existence of all d-regular nut graphs of higher orders is established. For even d the orders n are indeed consecutive, while for odd d the orders n are consecutive even numbers. Furthermore, necessary conditions for combinations of order and degree for vertex-transitive nut graphs are derived.